Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Felix Hausdorff wikipedia , lookup
Michael Atiyah wikipedia , lookup
Surface (topology) wikipedia , lookup
Sheaf (mathematics) wikipedia , lookup
Geometrization conjecture wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Fundamental group wikipedia , lookup
Covering space wikipedia , lookup
Grothendieck topology wikipedia , lookup
(December 25, 2014) Unwinding and integration on quotients Paul Garrett [email protected] http://www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/˜garrett/m/mfms/notes 2013-14/11a integration on quots.pdf] 1. 2. 3. 4. 5. 6. Surjectivity of averaging maps Invariant measures and integrals on quotients H\G Uniqueness of invariant integrals Preview of vector-valued integrals Mapping property of Gelfand-Pettis integrals Appendix: apocryphal lemma X ≈ G/Gx and other background The simplest case of unwinding is for f ∈ Cco (R): Z X R/Z Z f (x + n) dx = f (x) dx R n∈Z In fact, the integral on the quotient R/Z is unequivocally characterized [1] by this relation, once we know P that the averaged functions n f (x + n) are at least dense in C o (R/Z). As corollary, for F ∈ C o (R/Z), since F · f ∈ Cco (R), Z F (x) X R /Z Z f (x + n) dx = R/Z n∈Z Z X = R/Z X F (x) f (x + n) dx n∈Z Z F (x + n) f (x + n) dx = F (x) f (x) dx R n∈Z We need analogous assertions with less elementary group actions and less transparent representatives for the quotients. For example, with Γ = SL2 (Z) and H the upper half-plane, integration on Γ\H is characterized by requiring, for all f ∈ Cco (H), Z Γ\H X γ∈Γ f (γz) Z dx dy dx dy = f (z) 2 y y2 H P once we know that the averages γ∈Γ f (γz) are at least dense in Cco (Γ\H). In fact, such averaging maps are universally surjective on compactly-supported continuous functions, as demonstrated just below. An important variant [2] uses f ∈ Cco (Γ∞ \H) for a subgroup Γ∞ of Γ. By the surjectivity of averaging maps, take ϕ ∈ Cco (H) such that X ϕ◦β = f β∈Γ∞ [1] The Riesz-Markov-Kakutani theorem asserts that every (continuous) functional on compactly-supported R continuous functions on a reasonable topological space X is f → X f (x) dµ(x) for some measure µ. Relying on this, specification of a functional (integration) on Cco (X) specifies a measure. In fact, we care more about the integral than about the measure. 2 [2] This variant of unwinding arose most prominently in the Rankin-Selberg method, where R Γ\H |f | ·Es for cuspform f and Eisenstein series Es is unwound using the definition of Es as wound up from y s . This theme is pervasive in the theory of automorphic forms. 1 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) so then Z Γ\H X f ◦γ Z = Γ\H γ∈Γ∞ \Γ Z γ∈Γ∞ \Γ Z = X X Γ∞ \H Z X = Γ\H β∈Γ∞ X ϕ = H (ϕ ◦ β) ◦ γ ϕ◦β ϕ◦γ γ∈Γ Z = f Γ∞ \H β∈Γ∞ The corollary with F ∈ C o (Γ\H) and f ∈ Cco (Γ∞ \H) is Z F· Γ\H X f ◦γ Z = Γ\H γ∈Γ∞ \Γ X (F · f ) ◦ γ Z F ·f = Γ∞ \H γ∈Γ∞ \Γ Letting L1 (G), L1 (Γ\H), and L1 (Γ∞ \H) be the completions of Cco (H), Cco (Γ\H), and Cco (Γ∞ \H) with respect to the corresponding L1 norms Z Z |f |L1 (H) = |f | Z |f |L1 (Γ\H) = |f | |f |L1 (Γ∞ \H) = |f | Γ∞ \H Γ\H H Extension by continuity gives the same unwinding property of integrals on L1 spaces. 1. Surjectivity of averaging maps By convention, a topological group is a locally compact, Hausdorff topological space G with a continuous group operation G × G → G, and continuous inversion map g → g −1 . To avoid pathologies with regard to measures on products, we require that topological groups have a countable basis. Let dg be a right G-invariant measure [3] on G, meaning that for f ∈ Cco (G) Z Z f (g) dg = G Z f (gh) dg = G f (g) d(gh−1 ) = G Z f (g) dg (for all h ∈ G) G Let δG : G → (0, +∞) be the modular function of G, gauging the discrepancy between left and right invariant −1 measures, in the sense that meas (gE) = δG (g) · meas (E) for a measurable set E ⊂ G. Then δG (g) dg is a left invariant measure. Let H be a closed subgroup of G, with right invariant measure dh, and modular function δH . [1.0.1] Lemma: The averaging map α : Cco (G) → Cco (H\G) by Z (αf )(g) = f (hg) dh H is surjective. Proof: Let q : G → H\G be the quotient map. Let U be a neighborhood of 1 ∈ G having compact closure U . For each g ∈ G, gU is a neighborhood of g. The images q(gU ) are open, by the characterization of the quotient topology. Given F ∈ Cco (H\G), the support spt(F ) of F is covered by the opens q(gU ), and admits a finite subcover q(g1 U ), . . . , q(gn U ). The set C = q −1 (spt(F )) ∩ g1 U ∪ . . . ∪ gn U ⊂ G [3] Right or left G-invariant positive regular Borel measures on G are (right or left) Haar measures on G. 2 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) is compact, and q(C) = spt(F ) ⊂ H\G. Let f ∈ Cco (G) be identically 1 on a compact neighborhood of C, and non-negative real-valued everywhere. Then αf ∈ Cco (H\G) is strictly positive on the compact set spt(F ), so has a strictly positive lower bound µ there. By continuity, V = {x ∈ H\G : αf (x) > µ/2} is an open subset of H\G containing q(C) = spt(F ), and αf (x) ≥ µ/2 on the closure V . Then F/αf is continuous on V . Let F (qx) αf (x) f (x) · Φ(x) = (for x 6∈ V ) 0 (for x ∈ V ) By design, α(Φ) = F , and Φ is continuous in V . This is the main argument. Continuity of Φ in the interior of the complement of V is clear, but one might worry about continuity of Φ at points on the boundary ∂V of V : Since V is compact, ∂V is compact. Thus, there is a neighborhood N of ∂V on which αf > µ/4. Every neighborhood of every point of ∂V contains a point not in V , so not in spt(F ), so by continuity f is 0 on ∂V . Given ε > 0, there is an open neighborhood N 0 of the compact set ∂V on which |F | < ε. On the neighborhood N 00 = N ∩ N 0 of ∂V the continuous quotient F/αf is bounded by 4ε/µ. Thus, assigning /// values 0 to Φ on ∂V is compatible both with values 0 off V and values F/αf in V . 2. Invariant measures and integrals on quotients H\G [2.0.1] Theorem: The quotient H\G has a right G-invariant measure if and only if δG H = δH . In that case, the integral is unique up to scalars, and is characterized as follows. For given right Haar measure dh on H and for given right Haar measure dg on G there is a unique invariant measure dġ on H\G such that for f ∈ Cco (G) Z Z Z f (hg) dh dg = f (g) dg (for f ∈ Cco (G)) H H\G G Proof: First, prove the necessity of the condition on the modular R functions. Suppose that there is a right G-invariant measure on H\G. Let α be the averaging map f → H f (hg) dh. For f ∈ Cco (G) the map Z f→ αf (ġ) dġ H\G momentarily emphasizing the coordinate ġ on the quotient, is a right G-invariant functional (with the continuity property as above), so by uniqueness of right invariant measure on G must be a constant multiple of the Haar integral Z f −→ f (g) dg G The averaging map behaves in a straightforward manner under left translation Lh f (g) = f (h−1 g) for h ∈ H: for f ∈ Cco (G) and for h ∈ H Z α(Lh f )(g) = −1 f (h Z xg) dx = δH (h) H by replacing x by hx. Then Z Z f (g) dg = G H\G α(f )(g) dġ = δ(h)−1 f (xg) dx H Z H\G 3 α(Lh f )(g) dġ = δ(h)−1 Z G f (h−1 g) dg Paul Garrett: Unwinding and integration on quotients (December 25, 2014) by comparing the iterated integral to the single integral. Replacing g by hg in the integral gives Z Z f (g) dg = δ(h)−1 δG (h) f (g) dg G G Choosing f such that the integral is not 0 implies the stated condition on the modular functions. Proof of sufficiency starts from existence of Haar measures on G and on H. For simplicity, first suppose that both groups are unimodular. As expected, attempt to define an integral on Cco (H\G) by Z Z αf (ġ) dġ = f (g) dg H\G G Cco (G) invoking the fact that the averaging map α to Cco (H\G) is surjective. The potential problem is R from well-definedness. It suffices to prove that G f (g) dg = 0 for αf = 0. Indeed, for αf = 0, for all F ∈ Cco (G), the integral of F against αf is certainly 0. Rearrange Z Z Z Z Z 0 = F (g) αf (g) dg = F (g) f (hg) dh dg = F (h−1 g) f (g) dg dh G −1 by replacing g by h G −1 g. Replace h by h H H G , so Z 0 = αF (g) f (g) dg G Surjectivity of α shows that F can be chosen so that αF is identically 1 on the support of f . Then the integral of f is 0, as claimed, proving the well-definedness for unimodular H and G. For not-necessarily-unimodular H and G, in the previous argument the left translation by h−1 produces a factor of δG (h−1 ). Then replacing h by h−1 converts right Haar measure to left Haar measure, so produces a factor of δH (h)−1 , and the other factor becomes δG (h). If δG (h) · δH (h)−1 = 1, then the product of these two factors is 1, and the same argument goes through, proving well-definedness. /// 3. Uniqueness of invariant integrals The uniqueness of invariant measure and integrals on a topological group G is a special case of a more general uniqueness result for invariant functionals. The general existence argument is of a different nature, but in tangible circumstances existence is often clear for other reasons. The argument here illustrates the usefulness of the Gelfand-Pettis or weak integral, itself discussed further below. A translation-invariant function f on the real line, that is, a function with f (x + y) = f (x) for all x, y ∈ R, is constant, by a point-wise argument: f (x) = (Tx f )(0) = f (0) (with translation action Tx f (y) = f (x + y)) The same conclusion holds for translation-invariant distributions, but we cannot argue in terms of point-wise values. [3.0.1] Theorem: (Uniqueness of Haar measure) On a topological group G, [4] there is a unique right G-invariant element of the dual space Cco (G)∗ (up to constant multiples), namely the right-invariant integral Z f −→ f (g) dg (with right translation-invariant measure) G The same proof gives a much broader result: [4] A topological group is usually understood to be locally compact and Hausdorff. To avoid measure-theoretic pathologies, a countable basis is often assumed. Perhaps oddly, the local compactness excludes most topological vector spaces. 4 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) [3.0.2] Theorem: Let V ⊂ Cco (G) be a quasi-complete locally convex [5] topological vector space of complexvalued functions on G stable under left and right translations, and containing an approximate identity {ϕi }. In this context, an approximate identity on R a topological group G is a sequence of non-negative, compactlysupported continuous functions ϕi with G ϕi = 1, whose supports shrink to {1G }, meaning that, for every neighborhood U of 1, there is io such that for all i ≥ io the support of ϕi is inside U . Urysohn’s lemma implies that Cco (G) contains an approximate identity. On topological groups G that are not unimodular, that is, on which a right Haar measure is not a left Haar measure, nevertheless d(h−1 ) for right Haar measure is a left Haar measure, so for an approximate identity {ϕi } for a fixed right Haar measure, {g → ϕi (g −1 )} is an approximate identity for left Haar measure. Then there is a unique right G-invariant element of the dual space V ∗ (up to constant multiples), namely Z f −→ f (g) dg (with right translation-invariant measure) G Proof: Let Rg f (y) = f (yg) be the right translation action of G on functions on G, and Lg fR(y) = f (g−1 y) left translation. Let ϕi be an approximate identity for a fixed right Haar measure dg. Since G ϕi (g) dg = 1 and ϕi is non-negative, ϕi (g) dg is a probability measure [6] on the (compact) support of ϕi . Thus, for any f ∈ V , we have a V -valued Gelfand-Pettis integral Z Rϕi f = ϕi (g) Rg f dg ∈ closure of convex hull of {ϕi (g)f : g ∈ G} ⊂ V G The assumption V ⊂ Cco (G) is understood to entail that right and left translation action of G on V are continuous, meaning that G × V → V by g × f → Rg f and g → Lg f are (jointly) continuous. By continuity, given a neighborhood N of 0 in V , Rϕi f ∈ f + N for all sufficiently large i. Letting ϕ̌i (g) = ϕi (g −1 ) so that {ϕ̌i } is an approximate identity for left Haar measure d(g −1 ), also Lph ˇ f → f. i For a right-invariant (continuous) functional u on the space of functions V , Z u(f ) = lim u g → i ϕ̌i (h) f (h−1 g) d(h−1 ) Z = lim u g → i G ˇ (h−1 ) f (hg) dh ph i G replacing h by h−1 . After further replacing h by hg −1 , then move the functional u inside the integral via the Gelfand-Pettis integral property, and invoke the invariance: Z Z −1 ˇ u(f ) = lim u g → phi (gh ) f (h) dh = lim u g →, ϕ̌i (gh−1 ) f (h) dh i i G Z = lim i G Z u g →, ϕ̌i (g) f (h) dh = lim uϕ̌i · f (h) dh i G G Apparently the limit exists, and gives the constant. /// [5] The class of quasi-complete, locally convex topological vectorspaces includes essentially all reasonable examples: Hilbert, Banach, and Fréchet spaces, as well as LF spaces, that is, strict colimits of Fréchet, such as Cco (R) and Co∞ (R). Also included are these spaces’ weak-star duals, and other spaces of mappings such as the strong operator topology on mappings between Hilbert spaces, in addition to the uniform operator topology. The general definition of quasi-completeness requires a bit more background on topological vector spaces. [6] As usual, a probability measure is simply a non-negative, regular Borel measure with total measure 1. 5 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) 4. Preview of vector-valued integrals Rather than constructing integrals as limits following [Bochner 1935], [Birkhoff 1935], et alia, we use the [Gelfand 1936]-[Pettis 1938] characterization of integrals. Existence of Gelfand-Pettis integrals is proven separately, for vector spaces with adequate completeness properties. [7] A useful sample application is broad justification of differentiation of an integral with respect to a parameter. Let V be a topological vectorspace. For a continuous V -valued function f on a measure space X a GelfandPettis integral of f is If ∈ V such that Z (for all λ ∈ V ∗ ) λ◦f λ(If ) = X When it exists and is unique, this vector If would be denoted by Z If = Z f = X f (x) dx X In contrast to construction of integrals as limits of Riemann sums, the Gelfand-Pettis characterization is a property no reasonable notion of integral would lack. Since this property is an irreducible minimum, this definition of integral is called a weak integral. Uniqueness of the integral is immediate when the dual V ∗ separates points, meaning that for v 6= v 0 in V there is λ ∈ V ∗ with λv 6= λv 0 . This separation property certainly holds for Hilbert spaces: the map λw = hw, v − v 0 i is a continuous linear functional and λ(v − v 0 ) 6= 0 gives λv 6= λv 0 . The separation property for Banach spaces and more generally is part of the Hahn-Banach theorem. [8] [9] For the rest of this discussion, all topological vector spaces are assumed locally convex. Similarly, linearity of f → If follows when V ∗ separates points. The remaining issue is existence. [10] We integrate nice functions: compactly-supported and continuous, on measure spaces with finite, positive, Borel measures. In this situation, all the C-valued integrals Z Z λ◦f = X λ f (x) dx X exist for elementary reasons, being integrals of compactly-supported C-valued continuous functions on a compact set with respect to a finite Borel measure. [7] Precisely, things work out fine for quasi-complete, locally convex topological vectorspaces. Again, this class includes Hilbert, Banach, Fréchet spaces, LF spaces such as Cco (R) and Co∞ (R)), these spaces’ weak-star duals, and spaces of mappings such as the strong operator topology on mappings between Hilbert spaces, in addition to the uniform operator topology. [8] Hahn-Banach holds for all locally convex topological vector spaces, that is, topological vector space with a local basis at 0 consisting of convex sets. This includes Fréchet spaces, strict colimits of Fréchet spaces such as Cco (R) or Cc∞ (R), dual spaces of these, and essentially every reasonable space. [9] Although every reasonable topological vector space is locally convex, we can construct topological vector spaces without this property, whose main utility is illustrating the possibility of failure of local convexity. [10] We require that the integral of a V -valued function be in the space V itself, rather than in a larger space containing V , such as a double dual V ∗∗ , for example. Some discussions of integration do allow integrals to exist in larger spaces. 6 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) The crucial requirement on V for existence turns out to be that the convex hull of a compact set has compact closure. It is not too hard to show that Hilbert, Banach, or Fréchet spaces have this property, because of their metric completeness. Non-metrizable spaces need a subtler notion of completeness, quasi-completeness, meaning that bounded Cauchy nets converge. [11] [4.0.1] Theorem: Let X be a compact Hausdorff topological space with a finite, positive, Borel measure. Let V be a locally convex topological vectorspace in which the closure of the convex hull of a compact set is compact. Then continuous compactly-supported V -valued functions f on X have Gelfand-Pettis integrals. Further, Z f ∈ meas (X) · closure of convex hull of f (X) (Proof later.) X [4.0.2] Remark: The conclusion that the integral of f lies in the closure of a convex hull, is the substitute for the estimate of a C-valued integral by the integral of its absolute value. 5. Mapping property of Gelfand-Pettis integrals The characterization of Gelfand-Pettis integrals immediately and easily yields useful applications, many of them fitting under the following umbrella. Let X be a compact, Hausdorff topological space with a positive, regular Borel measure. Let T : V → W be a continuous linear map of locally convex, quasi-complete topological vector spaces. [5.0.1] Corollary: For a continuous V -valued function f on X, T Z f Z T ◦f = X X Proof: The right-hand side is the Gelfand-Pettis integral of the continuous, compactly-supported W -valued function T ◦ f , while the left-hand side is the image under T of the Gelfand-Pettis integral of f . Since W ∗ separates points, the equality will follow from proving that Z Z µ T f = µ T ◦f (for all µ ∈ W ∗ ) X X ∗ Noting that µ ◦ T ∈ V , from the characterization of the Gelfand-Pettis integrals, Z Z Z Z Z µ T f = (µ ◦ T ) f = (µ ◦ T )f = µ(T ◦ f ) = µ T ◦f X X X X X as desired. /// [5.1] Example: differentiation under the integral For F ∈ C k [a, b] × [c, d] , we claim that the C k [c, d]-valued function f on [a, b] given by f (x)(y) = F (x, y) [11] In topological vectorspaces lacking countable local bases, quasi-completeness is more relevant than completeness. For example, the weak ∗-dual of an infinite-dimensional Hilbert space is never complete, but is always quasi-complete. This example is non-trivial. 7 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) is continuous as a C k [c, d]-valued function, when the latter is given the standard Banach-space structure sup |ϕ(i) (y)| |ϕ|C k = sup 0≤i≤k y∈[c,d] Indeed, the partial derivatives of F (up to order k) with respect to its second argument are continuous on the compact set [a, b] × [c, d], so are uniformly continuous. That is, given ε > 0, there is δ > 0 such that for all x ∈ [a, b] and all 0 ≤ i ≤ k, ∂i ∂i sup sup i F (x, y) − i F (x0 , y) < ε (for |x − x0 | < δ) ∂y 0≤i≤k y∈[c,d] ∂y This is exactly what is meant by saying that f is a continuous C k [c, d]-valued function. By design, T = is a continuous linear map C k [c, d] → C k−1 [c, d]. Then the Gelfand-Pettis mapping property gives Z b Z b Z b Z b ∂ ∂ F (x, y) dx = T f (x) dx = T f (x) dx = F (x, y) dx ∂y a ∂y a a a ∂ ∂y That is, the interchange of these two limit processes is legitimate for general reasons. 6. Appendix: apocryphal lemma X ≈ G/Gx and other background Along with other background, the main point of this appendix is to prove that, with mild hypotheses, a topological space X acted upon transitively by a topological group G is homeomorphic to the quotient G/Gx , where Gx is the isotropy group of a chosen point x in X. Not assuming a mastery of point-set topology, nor a mastery of ideas about topological groups, several basic ideas need development in the course of the proof. Everything here is completely standard and widely useful. The discussion includes a variant of the Baire Category Theorem [12] for locally compact Hausdorff spaces. [6.0.1] Remark: Ignoring the topology, that is, as sets, the bijection G/Gx ≈ X by g · Gx ↔ gx is easy to see. In contrast, the topological aspects are not trivial. Surprisingly, the topology of the group G completely determines the topology of the set X on which it acts. [6.0.2] Proposition: Let G be a locally compact, Hausdorff topological group [13] and X a locally compact Hausdorff topological space with a continuous transitive action of G upon X. [14] Suppose that G has a countable basis. [15] Let x be any fixed element of X, and Gx the isotropy group [16] The natural map G/Gx → X by gGx → gx [12] The more common form of the Baire Category Theorem asserts that a complete metric space is not a countable union of closed sets each containing no non-empty open set. [13] As expected, this means that G is a group and is a topological space, the group multiplication is a continuous map G × G → G, and inversion is continuous. The local compactness is the requirement that every point has an open neighborhood with compact closure. The Hausdorff requirement is that any two distinct points x 6= y have open neighborhoods U 3 x and V 3 y that are disjoint, that is, U ∩ V = φ. [14] As expected, continuity of the action means that G × X → X by g × x → gx is continuous. The transitivity means that for any x ∈ X the set of images of x by elements of G is the whole set X, that is, {gx : g ∈ G} = X. [15] That is, there is a countable collection B (the basis) of open sets in G such that any open set is a union of sets from the basis B. [16] As usual, the isotropy (sub-) group of x in G is the subgroup of group elements fixing x, namely, G := {g ∈ G : x gx = x}. 8 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) is a homeomorphism. Proof: We must do a little systematic development of the topology of topological groups in order to give a coherent argument. [6.0.3] Claim: In a locally compact Hausdorff space X, given an open neighborhood U of a point x, there is a neighborhood V of x with compact closure V and V ⊂ U . Proof: By local compactness, x has a neighborhood W with compact closure. Intersect U with W if necessary so that U has compact closure U . Note that the compactness of U implies that the boundary [17] ∂U of U is compact. Using the Hausdorff-ness, for each y ∈ ∂U let Wy be an open neighborhood of y and Vy an open neighborhood of x such that Wy ∩ Vy = φ. By compactness of ∂U , there is a finite list y1 , . . . , yn T of points on ∂U such that the sets Uyi cover ∂U . ThenS V = i Vyi is open and contains x. Its closure is contained in U and in the complement of the open set i Wyi , the latter containing ∂U . Thus, the closure V of V is contained in U . /// [6.0.4] Claim: The map gGx → gx is a continuous bijection of G/Gx to X. Proof: First, G × X → X by g × y → gy is continuous by definition of the continuity of the action. Thus, with fixed x ∈ X, the restriction to G × {x} → X is still continuous, so G → X by g → gx is continuous. The quotient topology on G/Gx is the unique topology on the set (of cosets) G/Gx such that any continuous G → Z constant on Gx cosets factors through the quotient map G → G/Gx . That is, we have a commutative diagram /Z G y< y y y G/Gx Thus, the induced map G/Gx → X by gGx → gx is continuous. /// [6.0.5] Remark: We need to show that gGx → gx is open to prove that it is a homeomorphism. [6.0.6] Claim: For a given point g ∈ G, every neighborhood of g is of the form gV for some neighborhood V of 1. Proof: First, again, G × G → G by g × g → gh is continuous, by assumption. Then, for fixed g ∈ G, the map h → gh is continuous on G, by restriction. And this map has a continuous inverse h → g −1 h. Thus, h → gh is a homeomorphism of G to itself. In particular, since 1 → g · 1 = g, neighborhoods of 1 are carried to neighborhoods of g, as claimed. /// [6.0.7] Claim: Given an open neighborhood U of 1 in G, there is an open neighborhood V of 1 such that V 2 ⊂ U , where V 2 = {gh : g, h ∈ V } Proof: The continuity of G × G → G assures that, given the neighborhood U of 1, the inverse image W of U under the multiplication G × G → G is open. Since G × G has the product topology, W contains an open of the form V1 × V2 for opens Vi containing 1. With V = V1 ∩ V2 , we have V 2 ⊂ V1 · V2 ⊂ U as desired. /// [17] As usual, the boundary of a set E in a topological space is the intersection E ∩ E c of the closure E of E and the closure E c of the complement E c of E. 9 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) [6.0.8] Remark: Similarly, but more simply, since inversion g → g−1 is continuous and is its own (continuous) inverse, for an open set V the image V −1 = {g −1 : g ∈ V } is open. Thus, for example, given a neighborhood V of 1, replacing V by V ∩ V −1 replaces V by a smaller symmetric neighborhood, meaning that the new V satisfies V −1 = V . The following result is not strictly necessary, but sheds some light on the nature of topological groups. [6.0.9] Claim: Given a set E in G, closure E = \ E·U U where U runs over open neighborhoods of 1. [18] Proof: A point g ∈ G is in the closure of E if and only if every neighborhood of g meets E. That is, from just above, every set gU meets E, for U an open neighborhood of 1. That is, g ∈ E · U −1 for every neighborhood U of 1. We have noted that inversion is a homeomorphism of G to itself (and sends 1 to 1), so the map U → U −1 is a bijection of the collection of neighborhoods of 1 to itself. Thus, g is in the closure of E if and only if g ∈ E · U for every open neighborhood U of 1, as claimed. /// [6.0.10] Remark: This allows us to give another proof, for topological groups, of the fact that, given a neighborhood U of 1 in G, there is a neighborhood V of 1 such that V ⊂ U . (We did prove this above for locally compact Hausdorff spaces generally.) Proof: First, from the continuity of G × G → G, there is V such that V · V ⊂ U . From the previous claim, V ⊂ V · V , so V ⊂ V · V ⊂ U , as claimed. /// [6.0.11] Remark: We can improve the conclusion of the previous remark using the local compactness of G, as follows. Given a neighborhood U of 1 in G, there is a neighborhood V of 1 such that V ⊂ U and V is compact. Indeed, local compactness means exactly that there is a local basis at 1 consisting of opens with compact closures. Thus, given V as in the previous remark, shrink V if necessary to have the compact closure property, and still V ⊂ V · V ⊂ U , as claimed. [6.0.12] Corollary: For an open subset U of G, given g ∈ U , there is a compact neighborhood V of 1 ∈ G such that gV 2 ⊂ U . Proof: The set g−1 U is an open containing 1, so there is an open W 3 1 such that W 2 ⊂ g−1 U . Using the previous claim and remark, there is a compact neighborhood V of 1 such that V ⊂ W . Then V 2 ⊂ W 2 ⊂ g −1 U , so gV 2 ⊂ U as desired. /// [6.0.13] Claim: S Given an open neighborhood V of 1, there is a countable list g1 , g2 , . . . of elements of G such that G = i gi V . Proof: To see this, first let U1 , U2 , . . . be a countable basis. For g ∈ G, by definition of a basis, gV = [ Ui Ui ⊂gV Thus, for each g ∈ G, there is an index j(g) such that g ∈ Uj(g) ⊂ gV . Do note that there are only countably many such indices. For each index i appearing as j(g), let gi be an element of G such that j(gi ) = i, that is, gi ∈ Uj(gi ) ⊂ gi · V [18] This characterization of the closure of a subset of a topological group is very different from anything that happens in general topological spaces. To find a related result we must look at more restricted classes of spaces, such as metric spaces. In a metric space X, the closure of a set E is the collection of all points x ∈ X such that, for every ε > 0, the point x is within ε of some point of E. 10 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) Then, for every g ∈ G there is an index i such that g ∈ Uj(g) = Uj(gi ) ⊂ gi · V This shows that the union of these gi · V is all of G. /// A subset E of a topological space is nowhere dense if its closure contains no (non-empty) open set. [19] [6.0.14] Claim: (Variant of Baire Category theorem) A locally compact Hausdorff topological space is not a countable union of nowhere dense sets. [20] Proof: Let Wn be closed sets containing no non-empty open subsets. Thus, any non-empty open U meets the complement of Wn , and U −Wn is a non-empty open. Let U1 be a non-empty open with compact closure, so U1 − W1 is non-empty open. From the discussion above, there is a non-empty open U2 whose closure is contained in U1 − W1 . Continuing inductively, there are non-empty open sets Un with compact closures such that Un−1 − Wn−1 ⊃ Ūn Certainly Ū1 ⊃ Ū2 ⊃ Ū3 ⊃ . . . Then Ūi 6= φ, by compactness. [21] [22] Yet this intersection fails to meet any Wn . In particular, it cannot be that the union of the Wn ’s is the whole space. /// T Now we can prove that G/Gx ≈ X, using the viewpoint we’ve set up. Given an open set U in G and g ∈ U , let V be a compact neighborhood of 1 such that gV 2 ⊂ U . Let S g1 , g2S , . . . be a countable set of points such that G = i gi V . Let Wn = gn V x ⊂ X. By the transitivity, X = i Wi . We observed at the beginning of this discussion that G → X by g → gx is continuous, so Wn is compact, being a continuous image of the compact set gn V . So Wn is closed since it is a compact subset of the Hausdorff space X. By the (variant) Baire category theorem, some Wm = gm V x contains a non-empty open set S of X. For h ∈ V so that gm hx ∈ S, −1 gx = g(gm h)−1 (gm h)x ∈ gh−1 gm S [19] The union of all open subsets of a given set is its interior. Thus, a set is nowhere dense if its closure has empty interior. [20] The more common verison of the Baire category theorem asserts the same conclusion for complete metric spaces. The argument is structurally identical. [21] In Hausdorff topological spaces X compact sets C are closed, proven as follows. Fixing x not in C, for each y ∈ C, there are opens Uy 3 x and Vy 3 y with U ∩ V = φ, by the Hausdorff-ness. The Uy ’s cover C, so there is a finite T subcover, Uy1 , . . . , Uyn , by compactness. The finite intersection Wx = i Vyi is open, contains x, and is disjoint from C. The union of all Wx ’s for x 6∈ C is open, and is exactly the complement of C, so C is closed. [22] The intersection of a nested sequence C ⊃ C ⊃ . . . of non-empty compact sets C in a Hausdorff space X is n 1 2 non-empty. Indeed, the complements Cnc = X − Cn are open (since compact sets are closed in Hausdorff spaces), and if the intersection were empty, then the union of the opens Cnc would cover C1 . By compactness of C1 , there is a finite subcollection C1c , . . . , Cnc covering C1 . But C1c ⊂ . . . ⊂ Cnc , and Cnc omits points in Cn , which is non-empty, contradiction. 11 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) Every group element y ∈ G acts by homeomorphisms of X to itself, since the continuous inverse is given by −1 y −1 . Thus, the image gh−1 gm S of the open set S is open in X. Continuing, −1 −1 gh−1 gm S ⊂ gh−1 gm gm V x ⊂ gh−1 V x ⊂ gV −1 · V x ⊂ U x Therefore, gx is an interior point of U x, for all g ∈ U . /// Bibliography [Birkhoff 1935] G. Birkhoff, Integration of functions with values in a Banach space, Trans. AMS 38 (1935), 357-378. [Bochner 1935] S. Bochner, Integration von Funktionen deren Werte die Elemente eines Vektorraumes sind, Fund. Math., vol. 20, 1935, pp. 262-276. [Gelfand 1936] I. M. Gelfand, Sur un lemme de la theorie des espaces lineaires, Comm. Inst. Sci. Math. de Kharkoff, no. 4, vol. 13, 1936, pp. 35-40. [Pettis 1938] B. J. Pettis, On integration in vector spaces, Trans. AMS, vol. 44, 1938, pp. 277-304. [Phillips 1940] R. S. Phillips, Integration in a convex linear topological space, Trans. AMS, vol. 47, 1940, pp. 114-145. [Weil 1949] A. Weil, L’integration sur les groupes topologiques et ses applications, Herman, Paris, 1940. 12